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The image presents a complex number \( z = 3 - 3i\sqrt{3} \) and asks to find \( \sqrt[4]{z^2} \), the fourth root of the square of \( z \).

Let's break it down:

1. **Step 1: Calculate \( z^2 \)**  
   We need to square the complex number \( z = 3 - 3i\sqrt{3} \).

   \[
   z^2 = (3 - 3i\sqrt{3})^2
   \]

   Expanding this expression:

   \[
   z^2 = 3^2 - 2(3)(3i\sqrt{3}) + (3i\sqrt{3})^2
   \]
   \[
   z^2 = 9 - 18i\sqrt{3} + (9)(-3)
   \]
   \[
   z^2 = 9 - 18i\sqrt{3} - 27
   \]
   \[
   z^2 = -18 - 18i\sqrt{3}
   \]

2. **Step 2: Find the fourth root of \( z^2 \)**  
   We now need to find \( \sqrt[4]{-18 - 18i\sqrt{3}} \). To do this, it's easier to convert the complex number into polar form.

   The complex number \( -18 - 18i\sqrt{3} \) can be written as \( r(\cos \theta + i \sin \theta) \), where:
   - \( r = \sqrt{(-18)^2 + (-18\sqrt{3})^2} = \sqrt{324 + 972} = \sqrt{1296} = 36 \),
   - The argument \( \theta \) can be calculated as \( \theta = \tan^{-1}\left(\frac{-18\sqrt{3}}{-18}\right) = \tan^{-1}(\sqrt{3}) = \frac{2\pi}{3} \).

   So, in polar form:

   \[
   z^2 = 36 \left( \cos\left(\frac{2\pi}{3}\right) + i \sin\left(\frac{2\pi}{3}\right) \right)
   \]

3. **Step 3: Apply the fourth root**  
   To find \( \sqrt[4]{z^2} \), we take the fourth root of both the modulus and the argument:
   - The modulus becomes \( \sqrt[4]{36} = \sqrt{6} \),
   - The argument becomes \( \frac{1}{4} \times \frac{2\pi}{3} = \frac{\pi}{6} \).

   Therefore, \( \sqrt[4]{z^2} \) in polar form is:

   \[
   \sqrt[4]{z^2} = \sqrt{6} \left( \cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right) \right)
   \]

   Simplifying:

   \[
   \sqrt[4]{z^2} = \sqrt{6} \left( \frac{\sqrt{3}}{2} + i \cdot \frac{1}{2} \right)
   \]
   \[
   \sqrt[4]{z^2} = \frac{\sqrt{18}}{2} + i\frac{\sqrt{6}}{2}
   \]

   Hence, \( \sqrt[4]{z^2} \) is:

   \[
   \boxed{\frac{\sqrt{18}}{2} + i \frac{\sqrt{6}}{2}}
   \]

Let me know if you would like a more detailed explanation or have any questions! 

Here are some related questions:
1. How do you find the square of a complex number geometrically?
2. How do you convert a complex number into polar form?
3. What is the process of extracting the \( n \)-th root of a complex number?
4. What is the significance of Euler's formula in complex number operations?
5. Can you generalize the process of finding the fourth root of any complex number?

**Tip:** When working with complex numbers, converting to polar form can simplify powers and roots considerably.

Find the fourth root of the square of the complex number z = 3 - 3i√3.

Learn how to find the fourth root of the square of the complex number z = 3 - 3i√3. This solution explores key concepts in complex numbers, including converting to polar form and applying De Moivre's theorem. Suitable for undergraduate math students.

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